Introduc corporate finance ch5

Valuation of Bonds and Stock• First Principles: – Value of financial securities = PV of expected future cash flows • To value bonds and stocks we need to: – Estimate future cash flows

Valuation of Bonds and Stock

• First Principles:

– Value of financial securities = PV of expected

future cash flows

• To value bonds and stocks we need to:

– Estimate future cash flows:

• Size (how much) and

• Timing (when)

– Discount future cash flows at an appropriate rate:

• The rate should be appropriate to the risk presented by

the security

5.1 Definition and Example of a Bond

• A bond is a legally binding agreement between a

borrower and a lender:

– Specifies the principal amount of the loan.

– Specifies the size and timing of the cash flows:

• In dollar terms (fixed-rate borrowing)

• As a formula (adjustable-rate borrowing)

5.1 Definition and Example of a Bond

• Consider a U.S government bond listed as 6 3/8 of

December 2009.

– The Par Value of the bond is $1,000.

– Coupon payments are made semi-annually (June 30 and

December 31 for this particular bond)

– Since the coupon rate is 6 3/8 the payment is $31.875.– On January 1, 2002 the size and timing of cash flows are:



02 / 1 / 1

875

31

$

02 / 30 / 6

875

31

$

02 / 31 / 12

875

31

$

09 / 30 / 6

875

031 ,

1

$

09 / 31 / 12

5.2 How to Value Bonds

• Identify the size and timing of cash flows.

• Discount at the correct discount rate.

– If you know the price of a bond and the size and

timing of cash flows, the yield to maturity is the

discount rate.

Pure Discount Bonds

Information needed for valuing pure discount bonds:

– Time to maturity (T) = Maturity date - today’s date– Face value (F)

– Discount rate (r)

T

r

F PV

) 1

Pure Discount Bonds: Example

Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%

11 174

$ )

06 1 (

000 ,

1

$ )

1

$

30

Level-Coupon Bonds

Information needed to value level-coupon bonds:

– Coupon payment dates and time to maturity (T) – Coupon payment (C) per period and Face value (F) – Discount rate

T

F r

r

C PV

) 1

( )

1 (

1 1

Value of a Level-coupon bond

= PV of coupon payment annuity + PV of face value

Level-Coupon Bonds: Example

Find the present value (as of January 1, 2002), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent

– On January 1, 2002 the size and timing of cash flows are:



02 / 1 / 1

875

31

$

02 / 30 / 6

875

31

$

02 / 31 / 12

875

31

$

09 / 30 / 6

875

031 ,

1

$

09 / 31 / 12

30 049 ,

1

$ )

025

1 (

000 ,

1

$ )

025

1 (

1 1

2 05

875

Bond Rates and Yields

 Suppose a $1,000 face value bond currently sells for

$932.90, pays an annual coupon of $70, and matures

in 10 years

 The coupon rate is the annual dollar coupon expressed

as a percentage of the face value:

Coupon rate = $ _ /$ _ = 7.0%

 The current yield is the annual coupon divided by the price:

Current yield = $ _ / _ = 7.5%

Bond Rates and Yields

 The yield to maturity is the rate that makes the price of

the bond just equal to the present value of its future cash flows

 How to find yield to maturity?

– Trial and error

– Approximation formula

5.3 Bond Concepts

1 Bond prices and market interest rates move in opposite

directions

2 When coupon rate = YTM, price = par value

When coupon rate > YTM, price > par value (premium bond)

When coupon rate < YTM, price < par value (discount bond)

3 A bond with longer maturity has higher relative (%) price

change than one with shorter maturity when interest rate (YTM) changes All other features are identical

4 A lower coupon bond has a higher relative price change

than a higher coupon bond when YTM changes All other features are identical

YTM and Bond Value

800 1000 1100 1200 1300

When the YTM = coupon, the

bond trades at par

When the YTM > coupon, the bond trades at a discount

Maturity and Bond Price Volatility

C

Consider two otherwise identical bonds

The long-maturity bond will have much more

volatility with respect to changes in the

Coupon Rate and Bond Price Volatility

Consider two otherwise identical bonds

The low-coupon bond will have much more

volatility with respect to changes in the

High Coupon Bond

Low Coupon Bond

Bond Example:

 Bond J has a 4% coupon and Bond K a 10% coupon Both have 10 years to maturity, make semiannual payments, and have 9% YTMs If market rates rise by 2%, what is the

percentage price change of these bonds? What if rates fall

by 2%?

McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc All rights reserved.

 Percentage changes in bond prices

Bond prices and market rates

The results above demonstrate that, all else equal, the price of the

lower-coupon bond changes more (in percentage terms) than the price

of the higher-coupon bond when market rates change.

5.4 The Present Value of Common Stocks

• Dividends versus Capital Gains

• Valuation of Different Types of Stocks

– Zero Growth – Constant Growth – Differential Growth

Case 1: Zero Growth

• Assume that dividends will remain at the same level

forever

r P

r r

r

P

Div

) 1

(

Div )

1 (

Div )

1 ( Div

0

3

3 2

2 1

1 0

=

+ +

+ +

+ +

• Since future cash flows are constant, the value of a zero

growth stock is the present value of a perpetuity:

Case 2: Constant Growth

) 1

( Div Div1 = 0 + g

Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:

g r

1

2 Div ( 1 ) Div ( 1 )

3 0

Case 3: Differential Growth

• Assume that dividends will grow at different

rates in the foreseeable future and then will grow at a constant rate thereafter.

• To value a Differential Growth Stock, we need

to:

– Estimate future dividends in the foreseeable future. – Estimate the future stock price when the stock

becomes a Constant Growth Stock (case 2).

– Compute the total present value of the estimated

future dividends and future stock price at the appropriate discount rate.

Case 3: Differential Growth

) (1

Div Div1 = 0 + g1

• Assume that dividends will grow at rate g1 for N years and grow at rate g2 thereafter

2 1 0

1 1

N N

N Div (1 g ) Div (1 g ) Div = −1 + 1 = 0 + 1

) (1

) (1

Div )

(1 Div

Case 3: Differential Growth

) (1

Div0 + g1

• Dividends will grow at rate g1 for N years and grow at rate g2 thereafter

2 1

0(1 ) Div + g

0

2

g g

g

N

N

+ +

Case 3: Differential Growth

We can value this as the sum of:

an N-year annuity growing at rate g1

r

C P

) 1

(

) 1

(

1

plus the discounted value of a perpetuity growing at rate

g2 that starts in year N+1

N B

r

g

r P

) 1

(

Div

2

1 N

Case 3: Differential Growth

To value a Differential Growth Stock, we can use

N T

T

r

g

r r

g g

r

C P

) 1

(

Div

) 1

(

) 1

(

1 N 1

A Differential Growth Example

A common stock just paid a dividend of $2 The dividend is expected to grow at 8% for 3 years, then

it will grow at 4% in perpetuity

What is the stock worth?

5.9 Stock Market Reporting

HI LO STOCKSYM DIV % PE 100s HI LO CLOSE CHG

as $19.06 in

Gap pays a dividend of 9 cents/share

Given the current price, the dividend yield is ½ %

Given the current price, the

PE ratio is 15

6,517,200 shares traded hands in the last day’s trading

Gap ended trading

at $19.25, down

$1.75 from yesterday’s close

5.9 Stock Market Reporting

HI LO STOCKSYM DIV % PE 100s HI LO CLOSE CHG

52.75 19.06 Gap Inc GPS 0.09 0.5 15 65172 20.50 19 19.25 -1.75

Gap Incorporated is having a tough year, trading near their

52-week low Imagine how you would feel if within the past year you had paid $52.75 for a share of Gap and now had a share worth $19.25! That 9-cent dividend wouldn’t go very far in making amends

Yesterday, Gap had another rough day in a rough year Gap

“opened the day down” beginning trading at $20.50, which was down from the previous close of $21.00 = $19.25 + $1.75

Looks like cargo pants aren’t the only things on sale at Gap

5.10 Summary and Conclusions

In this chapter, we used the time value of money formulae from

previous chapters to value bonds and stocks

1 The value of a zero-coupon bond is

2 The value of a perpetuity is

T

r

F PV

) 1

( +

=

r C

PV =

5.10 Summary and Conclusions (continued)

3 The value of a coupon bond is the sum of the PV of the

annuity of coupon payments plus the PV of the par value

at maturity

4 The yield to maturity (YTM) of a bond is that single rate

that discounts the payments on the bond to the purchase price

T

F r

r

C PV

) 1

( )

1 (

1 1

5.10 Summary and Conclusions (continued)

5 A stock can be valued by discounting its dividends There

are three cases:

1 Zero growth in dividends

2 Constant growth in dividends

3 Differential growth in dividends

r

P0 = Div

g r

T

r

g

r r

g g

r

C P

) 1

(

Div

) 1

(

) 1

(

1 N